So we’ve seen that the logistic equation
with r equals 4 shows the butterfly effect
sensitive dependence on initial conditions
two imperceptibly different starting points
can lead a big differences in the orbit later on
and we saw this by these of computers
I used the computer to iterate a function
or solve the differential equation
and we saw these diverging trajectories
so I argued that this makes these systems
even though they’re deterministic essentially unpredictable
because we need a possible accuracy in order to do long term
where often even medium term prediction
in addition to posing a challenge to our predictions of real processes
it also posses a challenge or puzzle
for how we think about the computer results themselves
since computers don’t store numbers to infinite precision
and that’s what we need in order to do
an accurate long term prediction on the computer
we need enormous precision in the starting point
and in all the numbers along the way
so given that computers make tiny round off errors
due to how they represents decimals
we might wonder can we trust the result of computer simulations at all.
and the answer turns out to be yes
and there is a nice result that illustrates this
known as the shadowing lemma.
so what I want to do in this video is present
the basic idea behind the shadowing lemma
partly so you feel little better about the computer results
but also because it’s just a neat result and a fun way
and another way to think about
what’s sensitive dependence on initial conditions
and chaotic dynamics mean.
so here is the idea behind shadowing
Let’s say we’re studying an iterated function
like the logistic equation
and we choose an initial condition and then we compute
we use a computer to compute an orbit
and we could plot it in a time series plot
just like I was having the computer do
and maybe it looks something like this
so the problem is that
the computer is making round off errors
it doesn’t have infinite precision
and so because of sensitive dependence on initial conditions
this computed orbit is not actually the true orbit
for the initial condition that we chose
so initially I’ll drive in blue
the true orbit with this initial condition
I start here it would be close
but then this is sensitive dependence on initial conditions
the true orbit might depart from the computed orbit
so I wanted the computer to tell me
the orbit with this initial condition
and what the computer told me
is showing here in this black curve
however this black curve the computed orbit is not
the true orbit for this initial condition
the computer makes small round off errors
in how it is storing decimal numbers
and so the computed orbit is not the same as a true orbit
this is another manifestation on the butterfly effect
so at this point we can ask if this black curve,
this black time series is not the true orbit, what is it?
does it have any meaning at all
or is it just garbage just some sort of randomness
and amazingly it turns out that
this computed orbit does have some meaning
it’s the true orbit for some other initial condition
so let me draw, picture of what that might look like
the risk of clouding this up
ok, so the black time series is what we computed.
the blue is the true orbit for this initial condition
the red is some other true orbit
so this black curve is indeed an orbit of the logistic equation
or whatever it is we’re studying
it just happens to not be the exact orbit
or the true orbit for initial condition that I thought
but it’s a true orbit for some other initial condition
so I can still interpret this black time series
as a trajectory of the logistic equation
it’s not garbage, it’s not nonsense.
it just happens to not be the exact trajectory
I maybe thought I was initially studying
so this says that numerical results
or at least iterated maps like the logistic equation
even though round off error precision
in the computer finite precision in the computer
and the butterfly effect means that this black computed orbit
it’s not true a faithful representation
of the orbit for this initial condition
it’s still faithful representation of an orbit of the logistic equation
so we would say that this computed orbit in black
shadows this red orbit
it might not be exactly this true orbit
but it’s arbitrarily close to some true orbit
and this result that this is true is know as the shadowing lemma
and a lemma in mathematics is a result that’s used as an intermediate step
to prove or demonstrate some other central or more important result
in any event this is a pretty famous result
and it’s known as the shadowing lemma
and the phenomenon is shadowing
shadowing is when a computed orbit which is in a sense wrong
due to the butterfly effect and finite precision
nevertheless shadows comes arbitrarily close to some other true orbit
so shadowing is I think it’s a strange phenomena
and it’s fun and interesting to think about
Let me give an analogy to illustrate this idea behind shadowing
so let’s say you asked me to draw a portrait of somebody
and as you know because you’re seeing my draw
I’m actually not very good at drawing at all
so I try to draw a true accurate portrait of
whomever you asked me to make a portrait of
but since I’m not very good
I make a little mistake around the eyes
and I make a little mistake around the mouth
and I make a little mistake around the nose
I make little mistake actually kind of all the time
because I’m just not very very good at drawing
so the result is I hand you a portrait of that I’ve drawn
and you would look at it
and you’d say that looks nothing like the person
you were supposed to be drawing
and I would say, yeah you’re right sorry
but in the shadowing lemma picture I’ve drawn
yes what I’ve drawn is not an accurate picture of the person
I was trying to draw or you wanted me to draw
but nevertheless what I’ve drawn is
an accurate portrait of somebody else
so I haven’t drawn an accurate picture
of your friend you wanted portrait of
but of the six or seven billion people in the world
I got this just very very close to right for somebody else
so that’s the idea behind the shadowing lemma
you ask the computer to not draw portrait
but make time series of the particular initial condition
and the computer makes little errors
because it just has finite precision
it’s certainly better at arithmetic than
I am at drawing but it still has finite precision
and so it hands you back not a portrait but a time series
and this time series is not exactly what you wanted
it’s not the true exact time series for the initial condition
but it’s a it is true or arbitrarily close to true
an arbitrarily close to true time series for some other initial condition
so maybe you will be a little bit disappointed
but nevertheless what the computer has given you
does say something true about the dynamical system you’re studying
it is arbitrarily close to a true orbit of the dynamical system
I can’t prove the shadowing lemma in this course
it’s a pretty technical result
and we just don’t have the mathematical machinery to do it
but let me say a few things that
maybe will make it seem at least a little bit more plausible
Ok, so let’s imagine that the dynamical system
we’re interested in is not a deterministic iterated function
but is a fair coin something that's random
that has an element of chance in it.
so that just every time you toss the coin
with an equal probability it comes up heads or tails
and then you asked me to say
ok what are the next five outcomes going to be
so that’s in a sense the orbit for this system
and I might say oh I think it’s going to be
heads heads tails tails tails
and then the system ran and it turns out I was wrong
not surprisingly it’s hard to predict
perfectly random fair coin processes
so anyway so I said heads heads tails tails tails
and I’m wrong
but then I could say yeah alright I’m wrong
but I’m not totally wrong
because if you wait long enough
you’re going to see heads heads tails tails tails for sure
because all possible outcomes of heads and tails are equally likely
so yeah I told you a wrong result in this particular instance
but it’s true in that you really will see heads heads tails tails tails
so I’m not as wrong as you think I am
ok so that’s not surprising
or very deep statement about a random process like tossing coins
what’s interesting is the same sort of argument holds
for this deterministic system an iterated function
so you asked me to predict the next five or the next fifth orbits
and I and my calculator on the computer give you an answer
and it turns out that that answer is wrong compared to the true orbit
but I could say yeah ok sorry I got it wrong
but I guarantee you that this if you wait long enough
you would see this orbit so I’ve given you
something that’s actually true
so in a sense if a system is random or mixed up enough
then errors made in trying to predict the system
in this context can still produce results
that could have been produced by the system
and maybe another way to think about this is
the things I’m calling errors are not errors
in the sense that I forget the rules of arithmetic
or think that you know I think that two plus two is five
they’re just you know they are very small imprecisions
and then it’s the butterfly effect that amplifies those imprecisions
so in effect that we have these tiny errors
but then the macroscopic manifestation of these errors
is result of the dynamics of the system itself
so in a sense that the errors or imprecisions
I’m not even sure what word to use
arise from the dynamics of the system
not from some horrible blender or external source
and so maybe it’s not surprising that the system induced errors
or imprecisions are nevertheless in some way true to the system
ok, so again this is certainly not a proof of the phenomena of shadowing
but maybe these remarks help to make
shadowing seem a little bit more plausible
and as maybe you could tell in the last couple minutes
there are different notions of randomness
that might be appearing here is a process random
if it’s made by a random process can a deterministic process
have a random outcome
so there’s some ideas that we want to unpack here
and think more carefully about what one means
when one says something is random
so this will be the topic what is randomness
and how do we think about it
in dynamical systems of the next set of lectures.